# Show unbiased OLS estimator and expression for variance of OLS estimator

Consider the usual linear mixed model: $$Y=X \beta+ZB+\epsilon$$ where Y and $$\epsilon$$ are $$n$$-dimensional random variables and $$B$$ is a $$q$$-dimensional random variable independent of $$\epsilon$$ so we have: $$B \sim N_q(0,\Sigma)$$ and $$\epsilon \sim N_n(0, \sigma^2 I_n)$$. The matrices $$X$$ and $$Z$$ are model matrices of dimensions $$n \times p$$ and $$n \times q$$ and $$\beta \in R^p$$.

Now we consider the ordinary least squares (OLS) estimator for: $$\tilde{\beta}=(X^TX)^{-1}X^TY.$$ But I think this is not the ML estimator in the linear mixed model. Now I have to show that $$\tilde{\beta}$$ is an unbiased estimator of $$\beta$$ and that $$\text{Var}(\tilde{\beta})=(X^T X)^{-1} X^T(Z \Sigma_{\theta}Z^T)X(X^TX)^{-1}+\sigma^2(X^TX)^{-1}.$$ I'm not sure how to show it but I think that $$\tilde{\beta}$$ is an unbiased estimator of $$\beta$$ if its expected value is equal to the true value of the parameter. But how can I show that? And what can I do to show the expression for $$\text{Var}(\tilde{\beta})$$? What do they mean with the $$\theta$$ in $$\Sigma_{\theta}$$? I hope anyone can help me?

• Did you delete an identical post and posted it anew? Or is this question different from the one posted earlier today? Commented Dec 12, 2021 at 19:24
• It's just the same, I just hope more helpers is online now or that more people will see the question beacuse I use this profile there is more active than the other profile Commented Dec 12, 2021 at 19:28
• You have found a way to game the system, but it is not really ethical... Commented Dec 12, 2021 at 20:33
• Is this a homework problem? Commented Dec 13, 2021 at 2:26

$$E(Y|X,Z) = X \beta$$ because $$B$$ and $$\varepsilon$$ have mean $$0$$, so $$E(\tilde{\beta}) = E((X^T X)^{-1} X^T Y) = (X^T X)^{-1} X^T X \beta = \beta$$ For the variance we have $$Var(Y | X,Z) = Z \Sigma_{\theta} Z^T + \sigma I_n$$ because $$B$$ and $$\varepsilon$$ are independent. Then use a similar idea to show the desired expression.
The $$\theta$$ in $$\Sigma_{\theta}$$ reflects the fact that the covariance of the random effects is unknown and is parameterized by $$\theta$$. I.e., we are trying to estimate $$\theta$$.